#94740

order	512
nilpotency class	6
isoclinism	isoclinic class of #94690
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#1415
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₄×C₃₂	C₂²
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

65 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 72 points:

(2,5)(3,7)(6,8)(9,10,11,17,23,36,48,33,13,19,24,37,51,63,57,46,30,44,55,66,59,72,71,68,58,67,69,61,29,42,50,35,15,21,26,39,53,65,70,62,31,45,56,60,28,41,49,34,14,20,25,38,52,64,54,43,27,40,47,32,12,18,22,16);(1,2,4,6)(3,8,7,5)(10,16)(11,22)(12,23)(13,27)(14,30)(17,18)(19,43)(20,46)(21,35)(24,54)(25,57)(26,50)(28,59)(29,53)(31,58)(32,36)(33,40)(34,44)(37,64)(38,63)(39,42)(41,66)(45,68)(47,48)(49,55)(51,52)(56,71)(60,72)(61,65)(62,67)(69,70)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=efg,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,c^b=cf,d^b=def,f^b=fg,g^b=gh,h^b=hi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1936);